In "A Modest Proposal," in paragraphs one through five, Swift uses the term, "mother." In paragraph six, he changes this to what word?
Swift's narrator in "A Modest Proposal" turns to the word "breeder" to describe mothers as he switches from discussing the plight of the poor in Ireland to his proposed solution to it. He uses the word mothers early on because he is trying to build our sympathy for the many poor Irish people who can't even afford to feed their children. He wants us to feel their pain and suffering.
However, when it comes to discussing his proposal, he becomes the hard-headed, bean-counting businessman, trying to convince us that his plan will turn a good profit. Suddenly, human beings he has felt such pity for turn into commodities—things—in his mind, and this change is reflected in his prose. It's almost as if, to be able to write his proposal, the narrator has to distance himself from what he is talking about, which is killing innocent human beings.
You are right in identifying that in paragraph six Swift cleverly uses a different word to describe the "mothers" that have gone before in previous paragraphs. Clearly the diction that Swift employs is key to his overall success in overtly presenting a serious and reasonable "proposal" whilst at the same time highlighting the shocking and dehumanising way in which many of his contemporaries viewed, treated and thought of the Irish. Note the example that you refer to:
The number of souls in Ireland being usually reckoned one million and a half, of these I calculate there may be about two hundred thousand couples whose wives are breeders, from which number I subtract thirty thousand couples, who are able to maintain their own children, although I apprehend there cannot be so many under the present distresses of the kingdom, but this being granted, there will remain an hundred and seventy thousand breeders.
The word "mothers" has been exchanged for "breeders," which clearly suggests a view where Irish women are dehumanised and viewed only for their ability to "breed" in the same way that animals are referred to. Note how Swift uses this term ironically; he is actually reinforcing the point that women are not animals but human beings, but highlighting the way in which they are being treated like animals at the moment.